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Although Desargues ' theorem chooses different roles for these ten lines and points, the Desargues configuration itself is more symmetric: any of the ten points may be chosen to be the center of perspectivity, and that choice determines which six points will be the vertices of triangles and which line will be the axis of perspectivity.
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Although and theorem
Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it.
Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic.
Although the proof is necessary to produce a theorem, it is not usually considered part of the theorem.
Although the virial theorem depends on averaging the total kinetic and potential energies, the presentation here postpones the averaging to the last step.
Although this at first appears to be a stronger statement, it is a direct consequence of the other form of the theorem, through the use of successive polynomial division by linear factors.
Although the issue of existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the Picard – Lindelöf theorem, that is far from the case for partial differential equations.
Although useful in its own right, the version of her theorem just given was a special case of the general version she derived in 1915.
Although there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them.
Although there is no proof that a smooth group order will be found in the Hasse-interval, by using heuristic probabilistic methods, the Canfield-Erdős-Pomerance theorem with suitably optimized parameter choices, and the L-notation, we can expect to try L < nowiki >/ 2, √ 2 < nowiki ></ nowiki > curves before getting a smooth group order.
Although he didn't publish much in the field, Dirichlet lectured on probability theory and least squares, introducing some original methods and results, in particular for limit theorems and an improvement of Laplace's method of approximation related to the central limit theorem.
Although that is strictly speaking a question about a real vector bundle ( the " hairs " on a ball are actually copies of the real line ), there are generalizations in which the hairs are complex ( see the example of the complex hairy ball theorem below ), or for 1-dimensional projective spaces over many other fields.
Although apparently a rather special case of the above theorem, the general result can be reduced to this simpler case.
Although in general these two can be different there is a theorem that, under very mild conditions, gives the equality of the two.
" Although this conjecture is true, most of its known proofs depend on the theory of separable and purely inseparable extensions ; for instance, in the case corresponding to the extension being separable, one known proof involves the use of the primitive element theorem in the context of Galois extensions.
Although Mordell's theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of every curve.
Although the deduction theorem could be taken as primitive rule of inference in such systems, this approach is not generally followed ; instead, the deduction theorem is obtained as an admissible rule using the other logical axioms and modus ponens.
Although Pappus's Theorem usually refers to Pappus's hexagon theorem, it may also refer to Pappus's centroid theorem.
Although the equipartition theorem makes very accurate predictions in certain conditions, it becomes inaccurate when quantum effects are significant, such as at low temperatures.
Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn – Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.
Although the full description of the splitting of every prime p in a general Galois extension is a major unsolved problem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primes p less than a large integer N, tends to a certain limit as N goes to infinity.
Although and chooses
Although Ohkawa chooses which projects they decide to decline or accept, Satsuki Igarashi decides on the actual time and order the group works on each project, creating the schedules for time allotted to each individual work.
Although an accused has the right to remain silent and may not be compelled to testify against himself, where an accused freely chooses to take the witness box and testify, there is no further right to silence and no general restriction on what kinds of questions they may be required to answer.
Although he could easily throw the batter-runner out at first base, the shortstop chooses to throw to the second baseman who is covering second base, in an attempt to force out the runner advancing from first.
Although the henchman chooses the red door, we hear him being attacked by a tiger anyway.
Although Beleg beseeches Túrin to return to Doriath, where he has been pardoned and would be welcomed, Túrin still pridefully refuses and Beleg loyally chooses to stay with Túrin.
Although the primary role is no longer one of litigation, the Attorney General still represents the Crown and government in court in some select, particularly important cases, and chooses the Treasury Counsel who handle most government legal cases.
However, " Nomad "' s true identity is Austin Ames ( Chad Michael Murray ), the popular, yet unhappy, quarterback of the school's football team, and Shelby's boyfriend ( Although Austin broke up with her, she chooses to ignore it ).
Although Jackson gains the upper hand, he chooses to gloat rather than finish Chong Li, who proceeds to brutally beat him, putting him in the hospital.
Although the character aids the loose formation of heroes that eventually become the Squadron Supreme, Nighthawk chooses to remain aloof and only interacts when necessary.
Although it later turns out that his father, Gol Kolea, is alive and in fact has joined the Ghosts himself, Kolea chooses to stay out of Dalin's life, feeling his return would be too much for the young boy to go through.
Although its official broadcast area is Cheltenham and Tewkesbury, it can be clearly received across a large part of both Gloucestershire and The Malverns and due to this fact the station chooses to service these areas.
Although the gift is often money, it can be anything agreed upon by bride and groom such as a house or viable business that is put in her name and can be run and owned entirely by her if she chooses.
Although she works officially for the President, she chooses on her own whether or not she wants to follow his orders.
Although WVCY holds a commercial license, the station chooses not to air advertising and asks for viewer support through donations via the station and VCY America Radio instead for funding.
Although Jackie claims nothing happened between herself and Kelso in Chicago, Hyde chooses to stay in the marriage instead of annulling, ending his and Jackie's relationship.
Although " IRB " is a generic term used by the FDA and HHS, each institution that establishes an IRB may use whatever name it chooses.
Although originally unsure of the meaning of Charlie's dying message, " Not Penny's Boat ," when the survivors split Sawyer chooses to go with Locke to the Barracks, telling Kate he was simply doing what he's always done: surviving.
Although Jennifer is shocked, the fact that Brad chooses to display interest in another girl at school makes her feel even more frustrated.
Although Defoe's protagonist ( Robinson Crusoe ) chooses to leave his home and set out into an unexplored and dangerous life abroad, Winkfield's protagonist's ( Unca Eliza Winkfield ) trials and adventures are forced upon her.
Although second in authority to the Amyrlin, the Keeper of the Chronicles has no real authority over Aes Sedai, except what the Amyrlin Seat chooses to give her.
Although it is a gambit, White cannot hold on to his extra pawn without making compromises in the deployment of his pieces, so he often chooses to return the pawn and retain the initiative.
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